I've seen this same problem posted for a number of different models. Aside from unplugging the earphones when not in use for those models (like yours) with an earphone jack built into the remote, I don't recall ever seeing a definitive solution.
Roku Streaming Stick +, 3810X, Ser YH0059427035, wifi - Samsung UN55ES6100
Roku Streaming Stick, 3600X, Ser 5S56D8240827, wifi - JVC EM37T
They're not going to roll a truck and look under the hood of your remote. Only remedy is a replacement remote, which to get they take off line to get private info on where to ship the remote. People get upset because they don't see a happy ending posted, that's because ungrateful people don't come back and give a positive feedback.
I use a RC-AL2 remote that has the same problem, fresh AAA batteries only last at most 8 weeks. I've noticed that when I view the remote with an IR sensitive camera, the IR LED will continue to flash even when no button is being pushed. Doesn't happen all the time, but has happened repeatedly. This would explain the excessive battery drain on this ROKU remote at least. It also means the remote has a design flaw built into the unit.
I am having an issue with rapid battery drain on my RC-AL2 remote as well that came with my streaming stick. All my other remotes that came with my other four roku devices work fine. Please advise how I can replace? Thank you!